Generalized Fibonacci sequences and their properties

Martin Bunder; Joseph Tonien

paper

Generalized Fibonacci sequences and their properties

arXiv:2106.15790

Abstract

Let $F_{n} (k)$ be the generalized Fibonacci number defined by (with $F_{i} (k)$ abbreviated to $F_{i}$ ): $F_{n} = F_{n - 1} + F_{n - 2} + \dots + F_{n - k}$ , for $n \geq k$ , and the initial values $(F_{0}, F_{1}, ..., F_{k - 1})$ . Let $B_{n} (k, j)$ be $F_{n} (k)$ with initial values given by $F_{j} = 1$ and, for $i < j$ and $j < i < k$ , $F_{i} = 0$ . This paper shows that any $F_{n} (k)$ can be expressed as the sum of $B_{n} (k, j)$ s. This paper also expresses $B_{n} (k, j)$ and $F_{n} (k)$ as finite sums, derives some properties and evaluates their 2-adic order for a range of values of $k, j$ and $n$ and those of $B_{n} (3, j)$ and $B_{n} (4, j)$ for most values of $j$ and $n$ .